Question: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{y^2 + 9y}{y^2 + 14y + 45}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{y^2 + 9y}{y^2 + 14y + 45} = \dfrac{(y)(y + 9)}{(y + 5)(y + 9)} $ Notice that the term $(y + 9)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(y + 9)$ gives: $r = \dfrac{y}{y + 5}$ Since we divided by $(y + 9)$, $y \neq -9$. $r = \dfrac{y}{y + 5}; \space y \neq -9$